Abstract
In this chapter we review Quillen’s work on rational homotopy theory. Quillen gives a sequence of rational homotopy categories and proves that they are all equivalent. To a simply connected space, he associates a simplicial set with trivial one-skeleton and then the simplicial loop group. From there he passes to simplicial (complete) Hopf algebra, then to a simplicial Lie algebra, to a differential graded Lie algebra, and finally to a differential graded co-algebra. The latter dualizes to a differential graded algebra homotopy equivalent to the p.l. forms on the space, so that Quillen’s construction agrees in homotopy theory with Sullivan’s.
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